Question:
Find the values of the sine, cosine, and tangent for ∠A
a. sin A =
, cos A =
, tan A =
b. sin A =
, cos A =
, tan A =
c. sin A =
, cos A =
, tan A =
d. sin A =
, cos A =
, tan A =
Answer:
d. sin A =
, cos A =
, tan A =
Explanation:
The triangle for the question has been attached to this response.
As shown in the triangle;
AC = 36ft
BC = 24ft
ACB = 90°
To calculate the values of the sine, cosine, and tangent of ∠A;
i. First calculate the value of the missing side AB.
Using Pythagoras' theorem;
⇒ (AB)² = (AC)² + (BC)²
Substitute the values of AC and BC
⇒ (AB)² = (36)² + (24)²
Solve for AB
⇒ (AB)² = 1296 + 576
⇒ (AB)² = 1872
⇒ AB =
⇒ AB =
ft
From the values of the sides, it can be noted that the side AB is the hypotenuse of the triangle since that is the longest side with a value of
ft (43.27ft).
ii. Calculate the sine of ∠A (i.e sin A)
The sine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the hypotenuse side of the triangle. i.e
sin Ф =
-------------(i)
In this case,
Ф = A
opposite = 24ft (This is the opposite side to angle A)
hypotenuse =
ft (This is the longest side of the triangle)
Substitute these values into equation (i) as follows;
sin A =
sin A =
Rationalize the result by multiplying both the numerator and denominator by
sin A =
sin A =
iii. Calculate the cosine of ∠A (i.e cos A)
The cosine of an angle (Ф) in a triangle is given by the ratio of the adjacent side to that angle to the hypotenuse side of the triangle. i.e
cos Ф =
-------------(ii)
In this case,
Ф = A
adjacent = 36ft (This is the adjecent side to angle A)
hypotenuse =
ft (This is the longest side of the triangle)
Substitute these values into equation (ii) as follows;
cos A =
cos A =
Rationalize the result by multiplying both the numerator and denominator by
cos A =
cos A =
iii. Calculate the tangent of ∠A (i.e tan A)
The cosine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the adjacent side of the triangle. i.e
tan Ф =
-------------(iii)
In this case,
Ф = A
opposite = 24 ft (This is the opposite side to angle A)
adjacent = 36 ft (This is the adjacent side to angle A)
Substitute these values into equation (iii) as follows;
tan A =
tan A =
